Is it possible to give a texture thickness?
Voxels

Consider some images where the planes (rectangles) are either parallel or pependicular .. based on cubes.  Flat Surface on Top.


Small Sample of Color Transparency Threshold





Small Sample of High Density Mesh from distorted image from user provided image from natural light camera. Image below.





Sample from artificial image with more precise edges. Image below.




Steps


Create a high density mesh with cubes.  Simple Cube (6 sides) with the 2 array modifiers, vertical Z (1) and horizontal X (1), repeat 40 or more on both, then apply both. (I suggest user use a high number 500 or more if possible)Vertex Remove Doubles. See image above.   The number 40 was for a quick test.
UV Project . Turn specularity to zero. Add your texture. Affect color and transparency.
Render


Observations


High Density Mesh. Sheet of cubes. Geometry modified. One mesh of many cubes internally. High Density Mesh is the same requirement for Displacement in texture panel. See below.
Please Improve. I used a visible margin for each cube in the original array modifier, in the example below.
You can have zero margin.  Thus some lines in interior.  
The more intricate the transparency the more need for density with this technique.
The texture below has .png transparency.  Careful image processing is necessary and the best image, or least offensive image, here can be improved.


The perceived thickness is determined by the alpha of the channel. Your can change transparent image or image sequence. with no further work. No particular modelling.  There are other ways to do this.  Quick example.  I have added an array modifier only to show you two views at once. Front and Side. Any undesirable details have not been investigated.



Displacement method.  Dense mesh from a plane subdivided. Texture affecting displacement.  There is no flat surface.  This technique is documented many places.  This may be the least suited to your purposes. Consider Subdivision Modifier. The image also shows the triangulated surface just for discussion purposes.



All the above should be adjusted to you tastes.

Optionally consider a particle system with the 


Texture determining [density].  
Texture determining [color] in emitter.